arX

iv:1

709.

0133

4v1

[cs.

CR

] 5

Sep

201

71

Optimal Power Allocation by Imperfect

Hardware Analysis in Untrusted Relaying

Networks

Ali Kuhestani, Student Member, IEEE, Abbas Mohammadi, Senior Member, IEEE,

Kai-Kit Wong, Fellow, IEEE, Phee Lep Yeoh, Member, IEEE,

Majid Moradikia, and Muhammad R. A. Khandaker, Member, IEEE

Abstract

By taking a variety of realistic hardware imperfections into consideration, we propose an optimal

power allocation (OPA) strategy to maximize the instantaneous secrecy rate of a cooperative wireless

network comprised of a source, a destination and an untrusted amplify-and-forward (AF) relay. We

assume that either the source or the destination is equipped with a large-scale multiple antennas

(LSMA) system, while the rest are equipped with a single antenna. To prevent the untrusted relay from

intercepting the source message, the destination sends an intended jamming noise to the relay, which is

referred to as destination-based cooperative jamming (DBCJ). Given this system model, novel closed-

form expressions are presented in the high signal-to-noise ratio (SNR) regime for the ergodic secrecy

rate (ESR) and the secrecy outage probability (SOP). We further improve the secrecy performance

of the system by optimizing the associated hardware design. The results reveal that by beneficially

distributing the tolerable hardware imperfections across the transmission and reception radio-frequency

(RF) front ends of each node, the system’s secrecy rate may be improved. The engineering insight is

that equally sharing the total imperfections at the relay between the transmitter and the receiver provides

the best secrecy performance. Numerical results illustrate that the proposed OPA together with the most

appropriate hardware design significantly increases the secrecy rate.

A. Kuhestani and A. Mohammadi are with the Electrical Engineering Department, Amirkabir University of Technology,

Tehran, Iran.

K.-K Wong and M. R. A. Khandaker are with the Department of Electronic and Electrical Engineering, University College

London, London WC1E 6BT, U.K.

P. L. Yeoh is with the School of Electrical and Information Engineering, The University of Sydney, NSW, Australia.

M. Moradikia is with the Electrical Engineering Department, Shiraz University of Technology, Shiraz, Iran.

2

Index Terms

Physical layer security, Untrusted relay, Hardware imperfections, Optimal power allocation, Hard-

ware design

I. INTRODUCTION

Security in wireless communication networks is conventionally implemented above the phys-

ical layer using key based cryptography [1]. To complement these highly complex schemes,

wireless transmitters can also be validated at the physical layer by exploiting the dynamic

characteristics of the associated communication links [2], [3]. Physical layer security (PLS) is

a promising paradigm for safeguarding fifth-generation (5G) wireless communication networks

without incurring additional security overhead [3].

Massive multiple-input multiple-output (MIMO) systems as a key enabling technology of

5G wireless communication networks provide significant performance gains in terms of spectral

efficiency and energy efficiency [4], [5]. This new technology employs coherent processing across

arrays of hundreds or even thousands of base station (BS) antennas and supports tens or hundreds

of mobile terminals [4]–[6]. As an additional advantage, massive MIMO is inherently more secure

than traditional MIMO systems, as the large-scale antenna array exploited at the transmitter can

precisely aim a narrow and directional information beam towards the intended receiver, such that

the received signal-to-noise ratio (SNR) is several orders of magnitude higher than that at any

incoherent passive eavesdropper [7]. However, these security benefits are severely hampered in

cooperative networks where the intended receivers may also be potential eavesdroppers [8]–[10].

In the context of PLS, cooperative jamming which involves the transmission of additional

jamming signals to degrade the received SNR at the potential eavesdropper can be applied by

source [9], [10], the intended receiver node [8] or a set of nodes, i.e., source and destination

or source and relay to beamform the jamming noise orthogonal to the spatial dimension of the

desired signal [11], [12]. Recently, several works have considered the more interesting scenario

of untrusted relaying [13]–[21] where the cooperative jamming is performed by the intended

receiver, which is referred to as destination-based cooperative jamming (DBCJ).

In real life, an untrusted, i.e., honest-but-curious, relay may collaborate to provide a reliable

communication. Several practical scenarios may include untrusted relay nodes, e.g., in ultra-dense

heterogenous wireless networks where low-cost intermediate nodes may be used to assist the

source-destination transmission. In these networks, it is important to protect the confidentiality

3

of information from the untrustworthy relay, while concurrently relying on it to increase the

reliability of communication. Thanks to the DBCJ strategy [8], positive secrecy rate can still

be attained in untrusted relay networks. In recent years, several works have focused on the

performance analysis [13]–[16], power allocation [17]–[20] and security enhancement [18], [21]

of untrusted relaying networks. To be specific, the authors in [17] proposed an optimal power

allocation (OPA) strategy to maximize the instantaneous secrecy rate of one-way relaying network

while two-way relaying scenario was considered in [18]. By exploiting the direct link, a source-

based artificial noise injection scheme was proposed in [19] to hinder the untrusted relay from

intercepting the confidential message. A power allocation strategy was also proposed in [19] to

optimally determine the information and jamming signal powers transmitted by the source. In

[20], the OPA problem with imperfect channel state information was investigated. Notably, all

the aforementioned works considered perfect hardware in the communication network.

In practice, hardware equipments experience detrimental impacts of phase noise, I/Q imbal-

ance, amplifier non-linearities, quantization errors, converters, mixers, filters and oscillators [22],

[23]. Each of the imperfections distorts the signals in its own way. While hardware imperfections

are unavoidable, the severity of the imperfections depends on the quality of the hardware used in

the radio-frequency (RF) transceivers. The non-ideal behavior of each component can be modeled

in detail for the purpose of designing compensation algorithms, but even after compensation there

remain residual transceiver imperfections [23]. This problem is more challenging especially in

high rate systems such as LTE-Advanced and 5G networks exploiting inexpensive equipments

[22]. Although most contributions in security based wireless networks have assumed perfect

transceiver hardware [8]–[21], or only investigated the impact of particular imperfections such

as I/Q imbalance [25] or phase noise [26] in the presence of an external eavesdropper, this paper

goes beyond these investigations by considering residual hardware imperfections in physical

layer security design [22]–[24].

In this paper, we take into account the OPA in a two-hop amplify-and-forward (AF) untrusted

relay network where all the nodes suffer from hardware imperfections and either the source or

the destination is equipped with large-scale multiple antennas (LSMA) [27], [28] while the other

nodes are equipped with a single antenna. The DBCJ protocol is operated in the first phase and

then the destination perfectly removes the jamming signal via self interference cancelation in

the second phase. For this system model, the main contributions of the paper are summarized

as follows:

4

• Inspired by [22]–[24], we first present the generalized system model for transceiver hardware

imperfections in our secure transmission network. Based on this, we calculate the received

instantaneous signal-to-noise-plus-distortion-ratio (SNDR) at the relay and destination.

• We formulate the OPA between the source and destination that maximizes the instantaneous

secrecy rate of untrusted relaying. Accordingly, novel closed-form solutions are derived for

the exact OPA. In addition, new simple solutions are derived for the OPA in the high SNR

regime.

• According to our OPA solutions, novel compact expressions are derived for the ergodic

secrecy rate (ESR) and secrecy outage probability (SOP) in the high signal-to-noise-ratio

(SNR) regime that can be applied to arbitrary channel fading distributions. To gain further

insights, new closed-form expressions are presented over Rayleigh fading channels. The

asymptotic results highlight the presence of a secrecy rate ceiling which is basically different

from the prefect hardware case. We highlight that this ceiling phenomenon is independent

of the fading characteristic of the two hops.

• We provide new insights for hardware design in DBCJ-based secure communications. To

this end, under the cost constraint of transceiver hardware at each node, we formulate the

hardware design problem for the aforementioned network to maximize the secrecy rate. The

results reveal that the secrecy rate can be improved by optimally distributing the level of

hardware imperfections between the transmit and receive RF front ends of each node.

Notation: We use bold lower case letters to denote vectors. IN and 0N×1 denote the Identity

matrix and the zeros matrix, respectively. ‖.‖, (.)H and (.)T denote the Euclidean norm, conjugate

transpose and transpose operators, respectively; Ex{·} stands for the expectation over the random

variable (r.v.) x; Pr(·) denotes the probability; fX(·) and FX(·) denote the probability density

function (pdf) and cumulative distribution function (cdf) of the r.v. X , respectively; the CN (µ, σ2)

denotes a circularly symmetric complex Gaussian RV with mean µ and variance σ2; diag(A)

stands for the main diagonal elements of matrix A; Ei(x) is the exponential integral [34, Eq.

(8.211)]. [·]+ = max{0, x} and max stands for the maximum value.

II. SIGNAL AND SYSTEM REPRESENTATION

A. System Model

As shown in Fig. 1, the system model under consideration is a wireless network with one

source (S), one destination (D) and one untrusted AF relay (R). While R is equipped with one

5

Fig. 1. Secure transmission under the presence of transceiver imperfections. The first and the second figures show the downlink

and the uplink transmissions, respectively. The relay acts as both helper and eavesdropper. The solid lines represent the first

phase of transmission while the dashed line represents the second phase of transmission.

antenna, S or D is equipped with LSMA denoted by Ns or Nd, respectively [29], [30]. This

corresponds to the downlink (DL) and uplink (UL) scenarios in a cellular system where the

base station is equipped with a LSMA and the mobile user and relay are equipped with a single

antenna [29]. In the DL scenario, S has many antennas, whereas D has a single antenna. The

opposite applies to the UL scenario.

All the nodes operate in a half-duplex mode. Accordingly, D cannot receive the transmitted

signal from S while transmitting the jamming signal and hence, the direct link between S and

D is unavailable. We also assume that the channels satisfy the reciprocity theorem [8]. In DL

transmission, the complex Gaussian channel from S to R and R to D are denoted by hsr ∼CN (0Ns×1, µsrINs) and hrd ∼ CN (0, µrd), respectively, and for UL transmission, they are denoted

by hsr ∼ CN (0, µsr) and hrd ∼ CN (0Nd×1, µrdINd), respectively. We consider slow fading such

that the channel coefficients vary independently from one frame to another frame and, they do not

change within one frame. The additive white noise ni (i ∈ {R,D}) at each receiver is represented

by a zero-mean complex Gaussian variable with variance N0. We define the SNRs per link as

6

γsr∆=ρ‖hsr‖2 and γrd

∆=ρ‖hrd‖2 and hence, the average SNRs per branch is given by γsr = ρµsr

and γrd = ρµrd, where ρ = PN0

represents the transmit SNR of the network. The maximum ratio

transmission (MRT) beamforming and maximal ratio combining (MRC) processing are applied

at the multi-antenna node to improve the overall system performance [29]. Let ν = γsrγrd

represent

the ratio between the source-to-relay and relay-to-destination SNRs. With LSMA, we consider

ν ≫ 1 in the DL scenario while ν ≪ 1 in the UL scenario. These two special cases are taken

into account in this paper. As observed in the numerical results, the analysis are satisfactory

even for moderate values of ν.

The DBCJ technique is applied to degrade the received signal at the untrusted relay such

that it cannot decipher the desired information. The whole transmission is performed based on

a time-division multiple-access (TDMA) based protocol such that the message transmission is

divided into two phases, i.e. the broadcast phase and the relaying phase. We consider a total

transmit power budget for S and D of P with power allocation factor λ ∈ (0, 1) such that the

transmit powers at S and D are λP and (1 − λ)P , respectively [14], [18]. As such, during the

first phase, while S transmits the intended signal with power λP , concurrently D jams with a

Gaussian noise to confuse the untrusted relay with power (1− λ)P . For simplicity, the transmit

power at R is set to P and accordingly, in the second phase of transmission, R simply broadcasts

the amplified version of the received signal with power of P .

In order to consider the residual transceiver imperfections (after conventional compensation

algorithms have been applied) at node i, i ∈ {S,R,D}, the generalized system model from

[22] is taken into account. The imperfection at transmission and reception segments denoted

by ηti and ηri respectively, are introduced as distortion noises. The experimental results in [23]

and many theoretical investigations in [23], [31] have verified that these distortion noises are

well-modeled as Gaussian distributions due to the central limit theorem. A key property is that

the variance of distortion noise at an antenna is proportional to the signal power at that antenna

[23]. Accordingly, for the DL scenario, we have [23]

ηtS∼CN

(0,

λPktS2

‖hsr‖2diag(|hsr1|2 ... |hsrNs

|2)),

ηtD∼CN(0, (1− λ)Pkt

D2),

ηrD∼CN(0, Pkr

D2|hrd|2

). (1)

7

Furthermore, we have

ηtS∼CN(0, λPkt

s2),

ηtD∼CN

(0,

(1− λ)PktD2

‖hrd‖2diag(|hrd1|2 ... |hrdNd

|2)),

ηrD∼CN

(0, Pkr

D2diag(|hrd1|2 ... |hrdNd

|2)), (2)

for the UL scenario. The imperfections at R of both cases are also given by

ηtR∼CN(0, Pkt

R2),

ηrR∼CN(0, Pkr

R2[λ‖hsr‖2 + (1− λ)‖hrd‖2

]), (3)

where the design parameters kti , k

ri > 0 for i ∈ {S,R,D} characterize the level of imperfections

in the transmitter and receiver hardware, respectively. These parameters can be interpreted as the

error vector magnitudes (EVMs). EVM determines the quality of RF transceivers and is defined

as the ratio of the average distortion magnitude to the average signal magnitude. Since the EVM

measures the joint effect of different hardware imperfections and compensation algorithms, it

can be measured directly in practice [22]. 3GPP LTE has EVM requirements in the range of

kti , k

ri ∈ [0.08, 0.175], where smaller values are needed to achieve higher spectral efficiencies

[24].

Remark 1 (Co-channel Interference): The analysis in this paper supports the scenario of

large enough number of interfering signals, which is typical in wireless environments where

the Gaussian assumption for the interference is valid by applying the central limit theorem [32].

B. Signal Representation

Let us denote xS and xD as the unit power information signal and the jamming signal, respec-

tively. According to the combined impact of hardware imperfections which is well-addressed

by a generalized channel model [22], the received signal at R for DL and UL scenarios can be

expressed, respectively as

yDLR =

(√λPwT

S xS + ηtST)hsr +

(√(1− λ)PxD + ηtD

T)hrd + ηrR + nR, (4)

and

yULR =

(√λPxS + ηtS

T)hsr +

(√(1− λ)PwT

DxD + ηtDT)hrd + ηrR + nR, (5)

8

where wS = hsrH

‖hsr‖ and wD = hrdH

‖hrd‖ represent the MRT transmit weight vectors at S and D,

respectively. Observe from (4) and (5) that the propagated distortion noises by S and D, and the

self-distortion noise at R are treated as interference at the untusted relay which is a potential

eavesdropper. As a result, the engineering insight is to beneficially forward these hardware

imperfections to make the system secure instead of injecting more artificial noise by S [9], [10],

[19], D [13]–[17], [20] or a friendly jammer [18].

Then the relay amplifies its received signal in the first phase by an amplification factor of 1

G =

√P

E|yR|2=

√ρ

AGλ+BG

, (6)

where AG = (γsr − γrd)(1 + krR2) + kt

S2γu − kt

D2γv and BG = γrd(1 + kr

R2) + kt

D2γv + 1. Note

that in the DL scenario γu = ρ∑ |hsri |4/‖hsr‖2 and γv = γrd, and in the UL scenario γu = γsr

and γv = ρ∑ |hrdi|4/‖hrd‖2. Then, the received signal at D for DL and UL scenarios after

self-interference (or jamming signal) cancelation are respectively, given by

yDLD =G

√λPwH

S hsrhrdxS︸ ︷︷ ︸Information signal

+GhrdnR + nD︸ ︷︷ ︸Noise

+GηtSThsrhrd +GηrRhrd +GηtD

Thrdhdr + ηtRhrd + ηrD︸ ︷︷ ︸

Distortion noise

,

(7)

and

yULD = G

√λPhsrhrdxS︸ ︷︷ ︸

Information signal

+GhrdnR + nD︸ ︷︷ ︸Noise

+GηtSThsrhrd +GηrRhrd +Gη

tDThrdhdr + ηtRhrd + η

rD︸ ︷︷ ︸

Distortion noise

.

(8)

According to (4) and (5) and after some algebraic manipulations, the SNDR at R is given by

γR =λν

ARλ+BR, (9)

where AR = krR2ν + kt

S2 γuγrd

− ktD2 γvγrd

− krR2 − 1 and BR = 1+ kr

R2 + kt

D2 γvγrd

+ 1γrd

. Based on (7)

and (8) and after some manipulations, the SNDR at D can be calculated as

γD =λγsr

ADλ+BD, (10)

where AD = (γsr−γrd)(krD2kr

R2+kr

R2kt

R2+k2

R+krD2)+γu(k

rD2kt

S2+kt

R2ktS2+kt

S2)+γv(k

rD2kt

D2−

1In our analysis, we assume that the EVMs are perfectly known and will consider estimation errors in our future work.

9

ktR2ktD2 − kt

D2) + (ν − 1)(1 + kr

R2) + γu

γrdktS2 − γv

γrdktD2

and BD = γrd(krR2kr

D2 + kr

R2kt

R2+ k2

R +

krD2) + γv(k

rD2kt

D2+ kt

R2ktD2+ kt

D2) + γv

γrdktD2+ 1

γrd+ k2

R + krD2 + 2. We define k2

R∆=kt

R2+ kr

R2

and k2D∆=kt

D2+ kr

D2 as the total imperfection level at R and D, respectively.

Remark 2 (Perfect Hardware): The received SNRs at R and D with perfect hardware were

derived in [13], [14], [17]. When setting the level of imperfections at the nodes to zero, the

derived SNDRs in this section reduce to the special case as follows [14]

γperfectR =

λγsr(1− λ)γrd + 1

and γperfectD =

λγsrγrdλγsr + (2− λ)γrd + 1

. (11)

As can be seen, the mathematical structure of the derived SNDRs in (9), (10) are more com-

plicated compared to the perfect hardware case in (11), since the terms γuγrd

and γuγrd

manifest in

the denominator. As such, it is non-trivial to propose an OPA solution for the general scenario

of imperfect hardware. This generalization is done in Section III and is a main contribution of

this work.

For DL scenario, (9) and (10) are simplified to

γR =aLλ

λ+ bLand γD =

cLλ

λ+ dL, (12)

where

aL =1

ξ1 − 1, bL =

τ1(ξ1 − 1)ν

, cL =γrd

τ2γrd + ξ1and dL =

τ3γrd + τ4ν(τ2γrd + ξ1)

, (13)

and, τ1 = 1+ krR2 + kt

D2, τ2 = kr

D2kr

R2 + kr

R2kt

R2+ k2

R + krD2, τ3 = τ2 + kt

D2krD2 + kt

R2ktD2+ kt

D2,

τ4 = 2 + k2R + k2

D and ξ1 = 1 + krR2. Moreover, for UL scenario, we obtain

γR =aSλ

1− λand γD =

bSλ

λ+ cS, (14)

where

aS =ν

ξ1, bS =

γsr(γsr − γrd)τ2 + (ν − 1)ξ1

and cS =τ2γrd + ξ2

(γsr − γrd)τ2 + (ν − 1)ξ1, (15)

and ξ2 = 2 + k2R + kr

D2. Based on (12) and (14), we can conclude that although the intercept

probability is reduced by increasing the imperfection at R, the secrecy rate is also degraded.

It is, therefore, of great interest to intelligently distribute the tolerable hardware imperfections

across the transmission and reception radio frequency (RF) front ends of R (and other nodes) to

improve the secrecy rate of the network. This hardware design approach is analyzed in Section

10

VI and is a main contribution of this paper.

III. OPTIMAL POWER ALLOCATION

This section proceeds to analyze the optimal power allocation problem with the aim of

maximizing the instantaneous secrecy rate. Extending the results in [17], [19], [20] where the

OPA was solved for perfect hardware, we investigate the power allocation factor λ under the

presence of hardware imperfections. To do so, the instantaneous secrecy rate is evaluated by [8]

Rs =1

2 ln 2

[ln(1 + γD)− ln(1 + γR)

]+. (16)

By substituting λ = 0 into (9), (10) and then (16), we find Rs = 0. Since our goal is to distribute

the power optimally between S and D, a non-negative secrecy rate is achievable. As such, the

instantaneous secrecy rate can be reformulated as

Rs =1

2 ln 2

[ln(1 + γD)− ln(1 + γR)

]. (17)

Given that log(·) is monotonically increasing, the maximization of Rs is equivalent to the

maximization of

φ(λ)∆=

1 + γD1 + γR

. (18)

Therefore, the OPA factor λ⋆ can be obtained by solving the following constrained optimization

problem

λ⋆ = arg max{φ(λ)

}

s.t. 0 < λ ≤ 1 (19)

Lemma 1: f(x) is a quasi-concave function in R, if and only if [33, Section 3.4.3]

∂f(x)

∂x= 0 ⇒ ∂2f(x)

∂x2≤ 0. (20)

Based on lemma 1, we have the following corollary.

Corollary 1: f(x) is a quasi-concave function in x ∈ [x1, x2], if∂f(x)∂x

|x=x1 > 0,∂f(x)∂x

|x=x2 < 0

and there is only one maximum over [x1, x2] (despite constant functions).

11

Proposition 1: φ(λ) is a quasiconcave function of λ in the feasible set 0 < λ ≤ 1 and the

optimal point is given by

λ⋆E =

bLdL(cL−aL)+√

−aLbLcLdL(bL−dL)(aLdL−bLcL−bL+dL)

aLbL(cL+1)−cLdL(aL+1); ν ≫ 1

1−√

aS(cS+1)(bS+cS+1)bScS

; ν ≪ 1(21)

Proof: The first-order derivative of φ(λ) on λ is given by

∂φ(λ)

∂λ=

ALλ2+BLλ+CL

[(aL+1)λ+bL]2[λ+dL]2; ν ≫ 1

ASλ2+BSλ+CS

[1+(aL−1)λ]2[λ+cL]2; ν ≪ 1

, (22)

where AL = −aLbL(cL+1)+cLdL(aL+1), BL = −2bLdL(aL−cL), CL = −aLbL dL2+bL

2cL dL,

AS = (−bL (aL − 1) cL − aL (bL + 1)), BS = −2 cL (aL + bL) and CS = −aLcL2+bLcL. As can

be seen from (22),∂φ(λ)∂λ

= 0 leads to two solutions on λ. It is easy to examine that the feasible

solution for practical values of kti and kr

i [22] are derived as (21). According to Corollary 1, we

find that φ(λ) is a quasiconcave function in the feasible set.

To make the further analysis tractable, we provide new compact expressions for the OPA in

the high SNR regime. In the case of DL, the expressions in (13) are simplified to

aL =1

ξ1 − 1, bL =

τ1(ξ1 − 1)ν

, cL =1

τ2, dL =

τ3τ2ν

, (23)

and in the case of UL scenario, the expressions in (15) are changed to

aS =ν

ξ1, bS =

ν

(ν − 1)τ2, cS =

1

ν − 1. (24)

By substituting (23) and (24) into (21), the OPA solutions in the high SNR regime can be

expressed in the following tractable forms

λ⋆High=

θLν

; DL

1− θSν ; UL(25)

where θL =√

τ3τ2(τ1 − τ3) +

τ3τ2(ξ1 − 1)− τ3 and θS =

√1+τ2ξ1

.

12

IV. ERGODIC SECRECY RATE

In this section, we derive the ESR of the proposed secure transmission scheme in each case

of DL and UL scenarios. Since it is not straightforward to obtain a closed-form expression for

the exact ESR of DL and UL scenarios (the exact ESR includes double integral expressions

due to the complicated structures of (21)), we therefore proceed by first deriving new analytical

expressions for the ESR in the high SNR regime that can be applied to arbitrary channel fading

distributions. Based on these, new closed-form expressions are derived for the ESR in Rayleigh

fading channels. Despite prior works in the literature [14], [16]–[21] that investigated the ESR

based on perfect hardware assumption in various untrusted relaying networks, we take into

account hardware imperfections. The new results in this section generalize the recent results in

[14], [17].

The ESR as a useful secrecy metric representing the rate below which any average secure

transmission rate is achievable [2]. As such, the ESR is given by

Rs = E

{Rs

}=

1

2 ln 2

[E

{ln(1 + γD)

}

︸ ︷︷ ︸T1

−E

{ln(1 + γR)

}

︸ ︷︷ ︸T2

]. (26)

In the following, we proceed to evaluate the parts T1 and T2 and then Rs for each case of DL

and UL scenarios. Towards this goal, we present the following useful lemma.

Lemma 2: For positive constants α1, α2 and α3, and non-negative r.v. Γ, the cdf of the new

r.v. Γ̂ = α1Γα2Γ+α3

is derived as

FΓ̂(x) =

FΓ

(α3x

α1−α2x

); 0 ≤ x < α1

α2

1 ; x ≥ α1

α2

(27)

Proof: We start from the definition of the cdf as follows

FΓ̂(x) = Pr{ α1Γ

α2Γ + α3≤ x

}= Pr

{Γ(α1 − α2x) ≤ α3x

}, (28)

where the last probability equals to one for α1 − α2x < 0. Otherwise, it equals to FΓ

(α3x

α1−α2x

).

13

1) Downlink Scenario: By plugging (25) into (12), we obtain

γR =θL

θL(ξ1 − 1) + τ1, (29)

γD =θLγrd

(τ2θL + τ3)γrd + ξ1θL + τ4. (30)

We find that all the terms in (29) and (30) are deterministic constants which leads to the secrecy

rate ceiling in the high SNR regime.

Based on lemma 2 and (30), the part T1 in (26) is given by

T1 = E

{ln(1 +

θLγrd(τ2θL + τ3)γrd + ξ1θL + τ4

)}=

∫ θLτ2θL+τ3

0

1− Fγrd

((ξ1θL+τ4)x

θL−(τ2θL+τ3)x

)

1 + xdx, (31)

where the last equation follows from the integration by parts. The expression in (31) is straight-

forwardly evaluated for any channel fading distribution, either directly or by a simple numerical

integration.

Furthermore, based on (30) the part T2 is a constant value as

T2 = ln(1 +

θLθL(ξ1 − 1) + τ1

). (32)

We conclude from (32) that the amount of information leakage is independent of the transmit

SNR and the position of the relay, and only depends on the EVMs at network nodes. By replacing

(31) and (32) into (26), the compact ESR expression is achieved for any channel distribution.

For the case of Rayleigh fading, due to the fact that γrd is an exponential r.v. and applying

[34, Eq. (4.337.2)], the part T1 can be expressed in a closed-form solution. By substituting this

and (32) into (26), the closed-form ESR expression becomes

RsDL

=1

2 ln 2

[e

1r2γrdEi(− 1

r2γrd

)− e1

r1γrdEi(− 1

r1γrd

)− ln(1 +

θLθL(ξ1 − 1) + τ1

)], (33)

where r1 = (1+τ2)θL+τ3ξ1θL+τ4

and r2 = τ2θL+τ3ξ1θL+τ4

. We conclude from (33) that the ESR is exclusively

characterized by the level of imperfections over nodes and γrd which is a function of the transmit

SNR and the distance-dependent channel gain µrd.

14

2) Uplink Scenario: Substituting (25) into (14) yields

γR=λ⋆Hν

(1− λ⋆H)ξ1

≈ 1√ξ1(1 + τ2)

, (34)

γD≈γsr

τ2(1 +√

1+τ2ξ1

)γsr + ξ2 − ξ1. (35)

Observe from (34) and (35) that only the first hop SNR, γsr contributes to the secrecy rate

performance. Following the similar procedure as the DL scenario, the ESR performance of the

UL case for arbitrary fading distribution can be expressed as

RsUL

=1

2 ln 2

(∫ 1

τ2(1+

√

1+τ2ξ1

)

0

1− Fγsr((ξ2−ξ1)x

1−τ2(1+√

1+τ2ξ1

)x)

1 + xdx− ln(1 +

1√ξ1(1 + τ2)

)). (36)

For the case of Rayleigh fading, the closed-form ESR expression is given by

RsUL

=1

2 ln 2

[e

1t2γsrEi(− 1

t2γsr

)− e1

t1γsr Ei(− 1

t1γsr

)− ln(1 +

1√ξ1(1 + τ2)

)], (37)

where t1 =1+τ2(1+

√1+τ2ξ1

)

ξ2−ξ1and t2 =

τ2(1+√

1+τ2ξ1

)

ξ2−ξ1. It is observed from (37) that the ESR is entirely

determined by the average channel gain of the first hop, the transmit SNR and the level of

imperfections of all the network nodes. We also find that increasing the number of antennas at

D has no impact on the ESR when Nd is large.

V. SECRECY OUTAGE PROBABILITY

In this section, similar to our ESR results, general expressions are first presented for the SOP

that can be applied to any channel distribution, under the presence of transceiver imperfections

and in the high SNR regime. Based on these, we derive novel closed-form expressions for the

SOP in Rayleigh fading channels.

The SOP denoted by Pso is a criterion that determines the fraction of fading realizations

where a secrecy rate Rt cannot be supported [13]. Accordingly, the overall SOP is defined as

the probability that a system with the instantaneous secrecy rate Rs is not able to support the

target transmission rate Rt; Pso = Pr{Rs < Rt

}.

In the following, we focus on each case of DL and UL, respectively.

15

1) Downlink Scenario: Substituting (30) into (17) and then based on the SOP definition we

obtain

PDLso = Pr

( θLγrd(τ2θL + τ3)γrd + ξ1θL + τ4

≤ R̃t

)

=

Fγrd

((ξ1θL+τ4)R̃t

θL−(τ2θL+τ3)R̃t

);Rt <

12log2

(1+

θLτ2θL+τ3

1+γR

)

1 ;Rt ≥ 12log2

(1+

θLτ2θL+τ3

1+γR

) (38)

where R̃t = 22Rt(1+γR)−1 and γR is in (29), and the last equation follows from using lemma 2.

It is worth pointing out that the SOP expressions in (38) allows the straightforward evaluation of

the SOP for any channel fading distribution by a simple numerical integration. We can conclude

from (38) that the SOP is always 1 for target transmission rates more than a threshold (which

only depends on the EVMs of the nodes). Interestingly, this event holds for any channel fading

distribution, any network topology and any transmit SNR. Therefore as explained in Section IV,

some secrecy rates can never be achieved due to secrecy rate ceiling. Furthermore, we conclude

that for target transmission rates smaller than the threshold, Pso approaches zero with increasing

SNR (similar to perfect hardware) whereas the SOP always equals one for target transmission

rates larger than the threshold. This result is fundamentally different to the perfect hardware

case where the SOP goes to zero with increasing SNR and for any target transmission rate [13],

[15], [17].

For Rayleigh fading channels, γrd is an exponential r.v. and therefore, our new and simple

closed-form SOP expression in the presence of transceiver hardware imperfection is given by

PDLso =

1− exp

(− (ξ1θL+τ4)R̃t

(θL−(τ2θL+τ3)R̃t)γrd

);Rt <

12log2

(1+

θLτ2θL+τ3

1+γR

)

1 ;Rt ≥ 12log2

(1+

θLτ2θL+τ3

1+γR

) (39)

We note that the results of this section generalize the results of [17] which were derived for the

case of untrusted relaying with perfect hardware.

2) Uplink Scenario: Similar to the DL case, the SOP can be obtained by substituting γD in

16

(35) into (17) and using the SOP definition. This yields

PULso =

Fγsr

((ξ2−ξ1)R̃t

1−τ2(1+√

1+τ2ξ1

)R̃t

);Rt <

12log2

(1+ 1

τ2(1+

√

1+τ2ξ1

)

1+γR

)

1 ;Rt ≥ 12log2

(1+ 1

τ2(1+

√

1+τ2ξ1

)

1+γR

)(40)

where R̃t = 22Rt(1+γR)−1 and γR is in (34). For the special case of Rayleigh fading channels,

the closed-form SOP is derived as

PULso =

1− exp(

−(ξ2−ξ1)R̃t

(1−τ2(1+√

1+τ2ξ1

)R̃t)γsr

);Rt <

12log2

(1+ 1

τ2(1+

√

1+τ2ξ1

)

1+γR

)

1 ;Rt ≥ 12log2

(1+ 1

τ2(1+

√

1+τ2ξ1

)

1+γR

)(41)

As observed in the numerical results, the closed-form expressions (39) and (41) are sufficiently

tight at medium and high transmit SNRs.

VI. HARDWARE DESIGN

In this section, we aim to gain some insights into the design of RF hardware to maximize the

secrecy rate in untrusted relaying networks. Depending on the fixed cost of each network node,

we show how the RF segments at the transmission and reception front ends can be designed.

Specially, given the maximum tolerable hardware imperfection of each node, we derive new

analytical results characterizing how the hardware imperfections should be distributed between

the transmission RF segment and the reception RF segment of each node to maximize the secrecy

rate. Therefore, we should find kti and kr

i , i ∈ {S,R,D} to maximize the secrecy rate such that

kti+kr

i = ktoti . Mathematically speaking, our goal is to solve the following optimization problem

(ktR, k

rR, k

tD, k

rD) = arg maxφ(λ⋆) (42)

s.t. ktR + kr

R = ktotR

ktD + kr

D = ktotD

Based on (30), (35) and (17), the instantaneous secrecy rate is an increasing function of the

transmit SNR. Since it is our aim to achieve high transmission rates, we consider the asymptotic

SNR regime ρ → ∞ [22] to solve the hardware design problem (42). As observed in numerical

studies, the results of the high SNR analysis can be applied successfully at finite SNRs.

17

In the asymptotic SNR regime and for any random distributions on γsr and γrd, the asymptotic

received SNDRs at R and D are respectively, given by

γ∞R =

θLθL(ξ1−1)+τ1

; DL

1√ξ1(1+τ2)

; UL, (43)

and

γ∞D =

θLτ2θL+τ3

; DL

1

τ2(1+√

1+τ2ξ1

); UL

. (44)

By substituting (43) and (44) into (18), the secrecy rate ceiling is given by

φ∞ =

((1+τ2)θL+τ3)((ξ1−1)θL+τ1)(ξ1θL+τ1)(τ2θL+τ3)

; DL√ξ1(τ2+1)(τ2+

√ξ1√τ2+1)

τ2(√ξ1+

√τ2+1)(

√ξ1√τ2+1+1)

; UL(45)

Some conclusions and insights can be concluded from (45). First, the secrecy rate ceiling

event appears in the asymptotic SNR regime, which significantly limits the performance of the

system. This event is different from the perfect hardware case, in which the ESR increases with

increasing SNR. Note that this ceiling effect is independent of the fading distribution.

In the following, we focus on the of DL and UL scenarios separately and then conclude about

the hardware design of the overall network.

A. Downlink Scenario:

In the following, we proceed to solve the optimization problem (42) by independently dis-

cussing on the hardware design at R and D as follows.

Proposition 2: Suppose ktR + kr

R = ktotR , hence the secrecy rate ceiling is maximized if kt

R =

krR =

ktotR

2.

Proof: Please see Appendix A.

Proposition 3: Suppose ktD + kr

D = ktotD , thus the secrecy rate ceiling is maximized if

ktD=

2k2R+2ktot

D2+3−

√4k4

R+8k2Rk

totD

2+4ktotD

4+12k2R−4ktot

D2+9

4ktotD

. (46)

18

Proof: Please see Appendix B.

B. Uplink Scenario:

Similar to DL scenario, two propositions are provided as follows.

Proposition 4: Suppose ktR + kr

R = ktotR , thus the secrecy rate ceiling is maximized if kt

R =

krR =

ktotR

2.

Proof: Please see Appendix C.

Proposition 5: Suppose ktD + kr

D = ktotD , hence the secrecy rate ceiling is a monotonically

decreasing function of krD.

Proof: In this case, we have

∂φ∞

∂krD

=∂φ∞

∂τ2

∂τ2∂kr

D

, (47)

where ∂τ2∂krD

= 2krR2kr

D+2krD > 0 and ∂φ∞

∂τ2in (67) is negative in the feasible set. As such, ∂φ∞

∂krD< 0.

Based on Propositions 2–5, we provide the following corollary as a conclusion of the analysis

which provides new insights into the system design.

Corollary 2: Consider a cooperative network in which one multiple antennas node communi-

cates with a single antenna node via a single antenna untrusted relay. Let us assume a predefined

cost can be assigned to each node. To maximize the secrecy rate of this network the following

considerations should be taken into account:

• According to Propositions 2 and 4, the total cost for the relay node should be divided by

half between the transmission and reception RF front ends, i,e., it is better to apply the same

level of imperfections at every transceiver chain, instead of utilizing a mix of high-quality

and low-quality transceiver chains.

• According to Proposition 5, to design the multiple antennas node, the designers are per-

suaded to use higher-quality hardware in reception RF front end and lower-quality hardware

19

0 5 10 15 20 25 30 35 40 450

0.5

1

1.5

2

2.5

3

3.5

4

Transmit SNR [dB]

Erg

odic

Sec

recy

Rat

e (b

its/

s/H

z)

LSMA−S, Exact OPA SimulationLSMA−D, Exact OPA SimulationLSMA−S, Approximate OPA using eq. (33)LSMA−D, Approximate OPA using eq. (37)LSMA−S, EPALSMA−D, EPASecrecy rate ceiling using eq. (45)

Perfect hardware

Imperfect hardware

Fig. 2. Ergodic secrecy rate versus transmit SNR for exact and the derived closed-form expressions under perfect and imperfect

transceiver hardwares. Number of antennas at source or destination is set to 16. For imperfect case with k = 0.1, the secrecy

rate ceiling is observed.

in the transmission RF front end, i.e, the hardware imperfections at the reception end of

the multiple antennas node should be close to zero.

• According to Proposition 3, to design the single antenna node, the quality of RF requirements

at the transmission end should obeys from (46). As observed in the numerical examples,

we obtain ktD >

ktotD

2for typical values of EVMs.

VII. NUMERICAL RESULTS AND DISCUSSIONS

In this section, numerical results are provided to verify the accuracy of the derived closed-

form expressions in Section IV and V for LSMA at S (LSMA-S) and LSMA at D (LSMA-D),

respectively, and also the cases of multiple antennas at S (MA-S) and multiple antennas at D

(MA-D). We compare our LSMA-based ESR performance with the exact ESR with Monte-Carlo

simulations where the OPA is numerically evaluated for finite numbers of antennas using the

bisection method. In addition, the equal power allocation (EPA) between S and D (i.e., λ = 0.5)

is plotted as a benchmark. Furthermore, the concepts of secrecy rate ceiling and the practical

hardware insights from Section VI are numerically presented. In our numerical evaluations, the

20

0 5 10 15 20 25 30 35 40 45

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

2.2

2.4

Transmit SNR [dB]

Erg

odic

Sec

recy

Rat

e (b

its/

s/H

z)

MA−S, Exact OPA SimulationMA−D, Exact OPA SimulationMA−S, Approximate OPA using eq. (33)MA−D, Approximate OPA using eq. (37)

Case of k=0.05

Case of k=0.1

Fig. 3. Ergodic secrecy rate versus transmit SNR for exact and the derived closed-form expressions under different levels of

hardware imperfections; k ∈ {0.05, 0.1}. Number of antennas at source or destination is set to 4.

transmission links between nodes are modeled by the Rayleigh fading channel and the average

channel gains are specified as µsr = µrd = 10. Moreover, for LSMA the number of antennas is

set to 16, and for MA the number of antennas is set to 4.

Fig. 2 depicts the ESR versus transmit SNR ρ in dB for both cases of DL and UL and for

perfect (k = ktR = kr

R = ktS = kt

D = krD = 0) and imperfect (k = 0.1) cases. The number of

antennas at S and D are set to Ns = Nd = 16. It is observed from the figure that the Monte-Carlo

simulation of the exact OPA evaluated using the bisection method is in good agreement with

the derived high SNR closed-form solutions in (33) and (37) for both perfect and imperfect

hardwares. In contrast to perfect hardware, the figure shows that the ESR ceiling phenomenon

occurs for imperfect hardware which reveals the performance limits of hardware-constrained

realistic networks in the high SNR regime. This figure also reveals that hardware imperfections

have low impact at low SNRs, but are significant in the high SNR regime. Furthermore, it is

observed that the proposed OPA increases the secrecy rate floor by approximately 1 bits/s/Hz

and 0.9 bits/s/Hz for DL and UL scenarios, respectively compared to the EPA (λ = 0.5).

In Fig. 3, we examine the accuracy of the derived closed-form solutions for MA-S and MA-D

by considering Ns = Nd = 4. As can be seen, the numerical and the theoretical curves are in

21

0 5 10 15 20 25 30 35 40

10−4

10−3

10−2

10−1

100

Transmit SNR [dB]

Secr

ecy

Out

age

Prob

abili

ty

Imperfect, Exact OPA Simulation

Perfect, Exact OPA Simulation

Imperfect, Approximate OPA using eq. (39)

Perfect, Approximate OPA using eq. (39)

Imperfect, EPA

Perfect, EPA

OPA, Rt=0.25 bits/s/Hz

EPA, Rt=0.25 bits/s/Hz

OPA, Rt=1 bits/s/Hz

OPA, Rt=2 bits/s/Hz

Fig. 4. Secrecy outage probability versus transmit SNR for DL transmission, with different target transmission rates and under

perfect (k = 0) and imperfect hardware (k = 0.1).

good agreement across all SNR regimes. Moreover, it is observed that by increasing the level

of hardware imperfections from k = 0.05 to k = 0.1, the achievable secrecy rate is degraded

approximately 1 bits/s/Hz in the high SNR regime.

Figs. 4 and 5 show the SOP as a function of the transmit SNR for LSMA based DL and UL

scenarios, respectively, and for different target secrecy rates. The theoretical curves were plotted

by the derived analytical expressions in (39) and (41) which are well-tight with the marker

symbols generated by the Monte-Carlo simulations. As observed from these figures, there is

only a negligible performance loss caused by transceiver hardware imperfections in the low

target secrecy rate of Rt = 0.25 bits/s/Hz, but by increasing the target secrecy rate to Rt = 1

bits/s/Hz or Rt = 2 bits/s/Hz, substantial performance loss is revealed. Interestingly, for Rt = 2

bits/s/Hz, the network with imperfect hardware is always in outage and secure communications is

unattainable-irrespective of the transmit SNR. This is exactly predicted by our analytical results

in section V. The reason is that this target secrecy rate is more than the derived thresholds

in (39) and (41), and as mentioned, the SOP of the system always equals one for Rt more

the thresholds. It can also be seen from the figures that despite the OPA technique that the

SOP curves with imperfect hardware and with perfect hardware have the same slope (and thus,

22

0 5 10 15 20 25 30 35 4010

−5

10−4

10−3

10−2

10−1

100

Transmit SNR [dB]

Secr

ecy

Out

age

Pro

babi

lity

Imperfect, Exact OPA Simulation

Perfect, Exact OPA Simulation

Perfect, Approximate OPA using eq. (41)

Perfect, Approximate OPA using eq. (41)

Imperfect, EPA

Perfect, EPA

OPA, Rt=1 bits/s/Hz

OPA, Rt=2 bits/s/Hz

OPA, Rt=0.25 bits/s/Hz

EPA, Rt=0.25 bits/s/Hz

Fig. 5. Secrecy outage probability versus transmit SNR for UL transmission, with different target transmission rates and under

perfect (k = 0) and imperfect hardware (k = 0.1).

hardware imperfections lead to only an SNR offset which is unveiled as a curve shifting to the

right), the SOP performance of the EPA technique approaches a non-zero saturation value in

the high SNR regime for imperfect hardware. This observation reveals the secrecy performance

advantage of the proposed OPA scheme compared with EPA.

Finally, we provide Figs. 6 and 7 to illustrate the insights for designing practical systems that

were presented in Section VI. In the simulation, we assume that the total hardware imperfection

over each node equals to 0.2, i.e., ktotR = ktot

D = 0.2. Based on Propositions 2 and 4, to maximize

the secrecy rate, we should design the transmission and reception RF front ends at R such that

ktR = kr

R = 0.1. For LSMA at S, based on Proposition 3, we obtain ktD = 0.13 and kr

D = 0.07

while for LSMA at D and based on Proposition 5, we should design the hardwares such that

ktD = 0.2 and kr

D = 0. By defining the hardware imperfection vector as IV = [ktR, k

rR, k

tD, k

rD],

we consider the following four different hardware design schemes:

• Design 1: R and D are designed randomly, for example IV= [0.15, 0.05, 0.1, 0.1],

• Design 2: R is designed optimally based on Propositions 2 and 4 while D is designed

randomly; IV= [0.1, 0.1, 0.1, 0.1],

• Design 3: R is designed randomly while D is designed optimally; For LSMA at S, IV=

23

0 5 10 15 20 25 30 35 40 450

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

Transmit SNR [dB]

Erg

odic

Sec

recy

Rat

e (b

its/s

/Hz)

OPA, Design 1OPA, Design 2OPA, Design 3OPA, Design 4EPA, Design 4

Fig. 6. Ergodic secrecy rate versus transmit SNR for LSMA at S. Various imperfection distributions over RF transmission and

reception ends are considered for ktotR = ktot

D = 0.2.

[0.15, 0.05, 0.13, 0.07] and for LSMA at D, IV= [0.15, 0.05, 0.2, 0], and

• Design 4: R and D are designed optimally; For LSMA at S, IV= [0.1, 0.1, 0.13, 0.07] and

for LSMA at D, IV= [0.1, 0.1, 0.2, 0].

The results depict that the hardware design 4 which is based on Propositions 2-5 provides

higher ESR performance compared to the case of random hardware design (Design 1) and

the cases of optimizing only one node (Designs 2 and 3). Furthermore, they show that the

analysis presented in Section VI (which was based on high SNR analysis), can be utilized

auspiciously at medium SNRs. In addition, as can be seen from these figures and mentioned

before, different hardware designs have the ESR performance close together at low SNR regime,

while the difference between the ESR performance of the designs is large at high SNR regime.

Finally, we can understand from the figure that the proposed OPA together with Design 4

significantly outperforms the scenario of EPA with Design 4.

VIII. CONCLUSION

Physical radio-frequency (RF) transceivers are inseparable segments in both traditional and

new emerging wireless networks. In the literature, very few works have considered the impact

24

0 5 10 15 20 25 30 35 40 450.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

Transmit SNR [dB]

Erg

odic

Sec

recy

Rat

e (b

its/

s/H

z)

OPA, Design 1OPA, Design 2OPA, Design 3OPA, Design 4EPA, Design 4

Fig. 7. Ergodic secrecy rate versus transmit SNR for LSMA at D. Various imperfection distributions over RF transmission and

reception ends are considered for ktotR = ktot

D = 0.2.

of hardware imperfections on security based transmissions and little is understood regarding

this impact on untrusted relaying networks. In this paper, by taking hardware imperfections

into consideration, we proposed an optimal power allocation (OPA) strategy to maximize the

instantaneous secrecy rate of a cooperative wireless network comprised of a source, a destination

and an untrusted amplify-and-forward (AF) relay. Based on our OPA solutions, new closed-form

expressions were derived for the ergodic secrecy rate (ESR) and secrecy outage probability (SOP)

with Rayleigh fading channels. The expressions effectively characterize the impact of hardware

imperfections and manifest the existence of a secrecy rate ceiling that cannot be enhanced by

increasing SNR or improving fading conditions. They also illustrate that hardware imperfections

have low impact at low SNRs, but are significant in the high SNR regime. This issue reveals that

hardware imperfections should be taken into account when developing high rate systems such as

LTE-Advanced and 5G networks. To improve the secrecy performance of the network, we finally

presented the hardware design approach. Numerical results depict that optimally distributing the

hardware imperfections between the transmission and reception RF segments can further improve

the secrecy performance.

25

APPENDIX A

Let take the first-order derivative of φ∞ on ktR using the chain rule in partial derivations as

follows

∂φ∞

∂ktR

=∂φ∞

∂θL

(∂θL∂τ1

∂τ1∂kt

R

+∂θL∂τ2

∂τ2∂kt

R

+∂θL∂τ3

∂τ3∂kt

R

+∂θL∂ξ1

∂ξ1∂kt

R

)

+∂φ∞

∂τ1

∂τ1∂kt

R

+∂φ∞

∂τ2

∂τ2∂kt

R

+∂φ∞

∂τ3

∂τ3∂kt

R

+∂φ∞

∂ξ1

∂ξ1∂kt

R

, (48)

where using (45), we obtain

∂φ∞

∂θL=

κ1θL2 + κ2θL + κ3

(θLξ1 + τ1)2(τ2θL + τ3)2, (49)

∂θL∂τ1

=τ3

2√τ2τ3 (τ1 − τ3)

, (50)

∂θL∂τ2

= −√

τ3(τ1 − τ3)

2τ2√τ2

− τ3(ξ1 − 1)

τ 22, (51)

∂θL∂τ3

=τ1 − 2τ3

2√τ2τ3(τ1 − τ3)

+ξ1 − 1

τ2− 1, (52)

∂θL∂ξ1

=τ3τ2, (53)

∂φ∞

∂τ1=

θL (τ2θL + τ3 + θL)

(θLξ1 + τ1)2 (τ2θL + τ3), (54)

∂φ∞

∂τ2= − θ2L(ξ1θL + τ1 − θL)

(τ2θL + τ3)2(ξ1θL + τ1), (55)

∂φ∞

∂τ3= − θL(ξ1θL + τ1 − θL)

(τ2θL + τ3)2(ξ1θL + τ1), (56)

∂φ∞

∂ξ1=

θ2L(τ2θL + τ3 + θL)

(ξ1θL + τ1)2(τ2θL + τ3), (57)

26

where κ1 = −τ1τ2(τ2 + 1) + τ3ξ1(ξ1 − 1), κ2 = −2τ1τ3(τ2 − ξ1 + 1) and κ3 = τ1τ3(τ1 − τ3). By

substituting krR = ktot

R − ktR into (45), we obtain

∂τ1∂kt

R

= −2ktotR + 2kt

R, (58)

∂τ2∂kt

R

= −2krD2(ktot

R − ktR)− 2(ktot

R − ktR)k

tR2+ 2(ktot

R − ktR)

2ktR + 4kt

R − 2ktotR , (59)

∂τ3∂kt

R

= −2krD2(ktot

R − ktR)− 2(ktot

R − ktR)k

tR2+ 2(ktot

R − ktR)

2ktR + 4kt

R − 2ktotR + 2kt

RktD2, (60)

∂ξ1∂kt

R

= −2ktotR + 2kt

R. (61)

Substituting (49)–(61) into (48) and after tedious manipulations yields

∂φ∞

∂ktR

=4(1− kr

D2)(ktot

R − 2ktR)(

4ktR2 − 4kt

RktotR + 2ktot

R2 + 2kr

D2 + kt

D2)2 . (62)

Expression (62) shows that φ∞ is a concave function of ktR in the feasible set and kt

R =ktotR

2is

the single solution to ∂φ∞

∂ktR= 0.

APPENDIX B

Following the similar approach in Proposition 2, we should evaluate ∂φ∞

∂ktD. Let substitute kr

D =

ktotD − kt

D into τ1, τ2, τ3 and then compute the following derivations

∂τ1∂kt

D

= 1,∂τ2∂kt

D

= −2krR2(ktot

D − ktD)− 2ktot

D + 2ktD, (63)

∂τ3∂kt

D

= −2krR2(ktot

D − ktD)− 2ktot

D + 2ktD + 2kt

D(ktR2+ 1) + 2kt

D(ktotD − kt

D)2 − 2kt

D2(ktot

D − ktD).

(64)

The expression ∂φ∞

∂ktDcan be obtained similar to (48) by changing kt

R to ktD. Then by substituting

(49)–(57) and (63), (64) into ∂φ∞

∂ktD, and after manipulations, we obtain

∂φ∞

∂ktD

= −2(2k2

RktD − 2kt

D2ktotD + 2kt

DktotD

2+ 3kt

D − 2ktotD

)

(2k2

R + 3ktD2 − 4kt

DktotD + 2ktot

D2)2 . (65)

It is straightforward to see that (65) is a concave function of ktD in the feasible set and the single

solution to ∂φ∞

∂ktD= 0 is simply calculated.

27

APPENDIX C

We can write

∂φ∞

∂krR

=∂φ∞

∂τ2

∂τ2∂kr

R

+∂φ∞

∂ξ1

∂ξ1∂kr

R

. (66)

Using (45) yields

∂φ∞

∂τ2= −

ξ1

[(2 (τ2+2) ξ1−τ2

2)√ξ1(1+τ2)+2ξ1(τ2(ξ1+1−τ2)+ξ1+1)

]

2τ22(√

ξ1√1+τ2+ 1

)2 (ξ1+

√ξ1√1+τ2

)2 , (67)

∂φ∞

∂ξ1=

(1 + τ2)[(τ2 + 2ξ1)

√ξ1(1 + τ2) + 2 (1 + τ2) ξ1

]

2τ2(√

ξ1√1 + τ2 + 1

)2 (ξ1 +

√ξ1√1 + τ2

)2 . (68)

Considering ktR = ktot

R − krR, one can obtain

∂τ2∂kr

R

= 4krR3 − 6kr

R2ktot

R + (2krD2 + 2ktot

R2+ 4)kr

R − 2ktotR , (69)

∂ξ1∂kr

R

= 2krR. (70)

By substituting (67)–(70) into (66) and solving ∂φ∞

∂krR= 0 yields kr

R =ktotR

2.

Acknowledgements

The authors would like to thank Prof. Lajos Hanzo for helpful comments to improve the paper.

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